At 10.30am, I thought it might have been a bit early for a wine tasting. But the eight people sitting in the cool stone tasting room at Bire Winery were already having a merry time of it. I had just cycled uphill from the seafront at Lumbarda, one of Kor?ula’s main wine villages, and had to stop myself from gulping down the proffered glass of chilled rosé. Fresh, pale and dry, it was the best I’d tasted outside of Provence.

It was an excellent prelude to the wine I had come to Croatia to taste: dry white grk, made from a grape grown nowhere else in the world. To add to its oddity, it’s female only, so it needs another grape to pollinate it. Step forward plavac mali, the full-bodied red that’s found around Dalmatia and also goes into Bire’s moreish rosé.

They’ve been making wine in Lumbarda for 2,000 years, and the wineries I was visiting were small concerns that have been in the family for anything from 200 to 600 years. From Bire I cycled past olive groves to Zure Winery, where Ivan and Marko Batisti? have developed a sparkling grk, the first on the island, along with a bold red (Rebellion) that blends plavac mali with four other varieties.

He also does a grk version of the Croatian dessert wine prošek, that’s a bit like Hungarian tokaj. All of these exquisite wines went beautifully with plates of octopus, local cheeses and smoked ham made from the family’s pigs – a tantalising taste of what is offered in Zure’s vine-covered restaurant.

One of the (non-sandy) beaches at Lumbarda. Photograph: Maremagnum/Getty Images

Grk thrives in the sandy soil of Lumbarda. And it’s sand, along with wine, that draws many to Lumbarda, as it has that rare thing on Croatia’s beautiful but rocky coast: sandy beaches. Two of them. Three if you count the narrow strip in the village opposite my hotel, the exceptionally friendly Hotel Borik. The hotel’s flower-filled terrace and its swimming pool were serene spots from which to take in views of the harbour. It was also within a short distance of the beaches and vineyards, whether you were on two wheels or two feet.

Bilin Žal and Pržina beaches top and tail one of the island’s narrowest points, with a sea of vineyards dividing the two. Personally, I preferred the pebbly Tatinja beach, where I could float in clear waters while paddleboarders and kayakers passed me on their way to the little islands of Vrnik and Gubavac.

Vineyards at Lumbarda. Photograph: Adam Batterbee

“I never go into the sea,” Stjepan Cebalo told me. His Popi? winery is one of the three biggest in Lumbarda, along with Bire and Zure (the village has seven). We sat on his covered terrace gazing at the sea, his vineyards, terracotta-roofed houses, the white spire of Sveti Rok church and the stark mountains and vineyards of the Pelješac peninsula beyond. The sea breeze was delicious on this 30C day. Like the Bire and Zure winery owners, Stjepan has sons who are happy to carry on the family business, bucking the trend of the rural desertification of other parts of Croatia. I was already wondering what on earth would make people want to leave this place.

Swapping my bike for a scooter, I nipped over to Kor?ula Town, 6.5km away. Its fishbone-shaped medieval old town was busy with day-trippers and people from small cruise ships and yachts. Many were swept along the Marco Polo trail that sprang up around the belief that the Venetian explorer was born in Kor?ula. I gave the Marco Polo Museum a miss and instead wandered through alleys of pale stone houses, stopping for a sublime lunch of seafood pasta on the waterfront terrace of the Leši? Dimitri Palace hotel.

Pupnatska Luka beach. Photograph: Simeone Huber/Getty Images

With motorised wheels, I headed into Kor?ula’s mountainous and thickly forested interior. As I got closer to the wine-growing villages of ?ara and Smokvica, the road swooped into valleys covered in vineyards of pošip (a more floral white wine than grk) and plavac mali. Kor?ula’s heavily indented southern coast shelters gorgeous little pebbly beaches, including a secluded one at Žitna and a larger, deeper bay at Pupnatska Luka.

Back in Lumbarda, the village was gearing up for its Friday fish festival, held weekly in summer. Fishermen and women set up grills in the square opposite the little village beach, and we all queued up for freshly grilled snapper, sardines and squid.

Someone lit a bonfire on the beach, and small children, teenagers and quite a few adults joined in a tradition of jumping over the fire. It was a glorious, joyful spectacle. Drinking plavac mali and eating seafood on the beach, I blessed Kor?ula’s intoxicating mix of water and wine.

•The trip was provided by Prestige Holidays, which has seven nights’ B&B at the Hotel Borik from £455pp, including flights to Dubrovnik and transfers. Scooter hire at Korkyra from €25 a day. For further information, visit croatia.hr